Question: Simplify and expand the following expression: $ \dfrac{a + 1}{2a - 10}-\dfrac{a}{a - 7} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(2a - 10)(a - 7)$ Multiply the first term by $\dfrac{a - 7}{a - 7}$ $ \begin{align*} \dfrac{a + 1}{2a - 10} \times \dfrac{a - 7}{a - 7} & = \dfrac{(a + 1)(a - 7)}{(2a - 10)(a - 7)} \\ & = \dfrac{a^2 - 6a - 7}{(2a - 10)(a - 7)}\end{align*} $ Multiply the second term by $\dfrac{2a - 10}{2a - 10}$ $ \begin{align*} \dfrac{a}{a - 7} \times \dfrac{2a - 10}{2a - 10} & = \dfrac{(a)(2a - 10)}{(a - 7)(2a - 10)} \\ & = \dfrac{2a^2 - 10a}{(a - 7)(2a - 10)}\end{align*} $ Now we have: $ = \dfrac{a^2 - 6a - 7}{(2a - 10)(a - 7)} - \dfrac{2a^2 - 10a}{(a - 7)(2a - 10)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{a^2 - 6a - 7 - (2a^2 - 10a)}{(2a - 10)(a - 7)} $ $ = \dfrac{a^2 - 6a - 7 - 2a^2 + 10a}{(2a - 10)(a - 7)} $ $ = \dfrac{-a^2 + 4a - 7}{(2a - 10)(a - 7)}$ Expand the denominator: $ = \dfrac{-a^2 + 4a - 7}{2a^2 - 24a + 70}$